Niven's theorem

Niven's theorem In mathematics, Niven's theorem, named after Ivan Niven, states that the only rational values of θ in the interval 0° ≤ θ ≤ 90° for which the sine of θ degrees is also a rational number are:[1] {displaystyle {begin{aligned}sin 0^{circ }&=0,\[10pt]sin 30^{circ }&={frac {1}{2}},\[10pt]sin 90^{circ }&=1.end{aligned}}} In radians, one would require that 0 ≤ x ≤ π/2, that x/π be rational, and that sin x be rational. The conclusion is then that the only such values are sin 0 = 0, sin π/6 = 1/2, and sin π/2 = 1.

The theorem appears as Corollary 3.12 in Niven's book on irrational numbers.[2] The theorem extends to the other trigonometric functions as well.[2] For rational values of θ, the only rational values of the sine or cosine are 0, ±1/2, and ±1; the only rational values of the secant or cosecant are ±1 and ±2; and the only rational values of the tangent or cotangent are 0 and ±1.[3] Contents 1 See also 2 References 3 Further reading 4 External links See also Pythagorean triples form right triangles where the trigonometric functions will always take rational values, though the acute angles are not rational Trigonometric functions Trigonometric number References ^ Schaumberger, Norman (1974). "A Classroom Theorem on Trigonometric Irrationalities". Two-Year College Mathematics Journal. 5 (1): 73–76. doi:10.2307/3026991. JSTOR 3026991. ^ Jump up to: a b Niven, Ivan (1956). Irrational Numbers. The Carus Mathematical Monographs. The Mathematical Association of America. p. 41. MR 0080123. ^ A proof for the cosine case appears as Lemma 12 in Bennett, Curtis D.; Glass, A. M. W.; Székely, Gábor J. (2004). "Fermat's last theorem for rational exponents". American Mathematical Monthly. 111 (4): 322–329. doi:10.2307/4145241. JSTOR 4145241. MR 2057186. Further reading Olmsted, J. M. H. (1945). "Rational values of trigonometric functions". The American Mathematical Monthly. 52 (9): 507–508. JSTOR 2304540. Lehmer, Derik H. (1933). "A note on trigonometric algebraic numbers". The American Mathematical Monthly. 40 (3): 165–166. doi:10.2307/2301023. JSTOR 2301023. Jahnel, Jörg (2010). "When is the (co)sine of a rational angle equal to a rational number?". arXiv:1006.2938 [math.HO]. External links Weisstein, Eric W. "Niven's Theorem". MathWorld. Niven's theorem at ProofWiki Categories: Rational numbersTrigonometryTheorems in geometryTheorems in algebra

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